(* Title: HOL/Tools/Transfer/transfer_bnf.ML
Author: Ondrej Kuncar, TU Muenchen
Setup for Transfer for types that are BNF.
*)
signature TRANSFER_BNF =
sig
val base_name_of_bnf: BNF_Def.bnf -> binding
val type_name_of_bnf: BNF_Def.bnf -> string
val lookup_defined_pred_data: Proof.context -> string -> Transfer.pred_data
val map_local_theory: (local_theory -> local_theory) -> theory -> theory
val bnf_only_type_ctr: (BNF_Def.bnf -> 'a -> 'a) -> BNF_Def.bnf -> 'a -> 'a
end
structure Transfer_BNF : TRANSFER_BNF =
struct
open BNF_Util
open BNF_Def
open BNF_FP_Util
open BNF_FP_Def_Sugar
(* util functions *)
fun base_name_of_bnf bnf = Binding.name (Binding.name_of (name_of_bnf bnf))
fun mk_Frees_free x Ts ctxt = Variable.variant_frees ctxt [] (mk_names (length Ts) x ~~ Ts) |> map Free
fun mk_Domainp P =
let
val PT = fastype_of P
val argT = hd (binder_types PT)
in
Const (@{const_name Domainp}, PT --> argT --> HOLogic.boolT) $ P
end
fun mk_pred pred_def args T =
let
val pred_name = pred_def |> prop_of |> HOLogic.dest_Trueprop |> fst o HOLogic.dest_eq
|> head_of |> fst o dest_Const
val argsT = map fastype_of args
in
list_comb (Const (pred_name, argsT ---> (T --> HOLogic.boolT)), args)
end
fun mk_eq_onp arg =
let
val argT = domain_type (fastype_of arg)
in
Const (@{const_name eq_onp}, (argT --> HOLogic.boolT) --> argT --> argT --> HOLogic.boolT)
$ arg
end
fun subst_conv thm =
Conv.top_sweep_conv (K (Conv.rewr_conv (safe_mk_meta_eq thm))) @{context}
fun type_name_of_bnf bnf = T_of_bnf bnf |> dest_Type |> fst
fun is_Type (Type _) = true
| is_Type _ = false
fun map_local_theory f = Named_Target.theory_init #> f #> Local_Theory.exit_global
fun bnf_only_type_ctr f bnf = if is_Type (T_of_bnf bnf) then f bnf else I
fun bnf_of_fp_sugar (fp_sugar:fp_sugar) = nth (#bnfs (#fp_res fp_sugar)) (#fp_res_index fp_sugar)
fun fp_sugar_only_type_ctr f fp_sugars =
(case filter (is_Type o T_of_bnf o bnf_of_fp_sugar) fp_sugars of
[] => I
| fp_sugars' => f fp_sugars')
(* relation constraints - bi_total & co. *)
fun mk_relation_constraint name arg =
(Const (name, fastype_of arg --> HOLogic.boolT)) $ arg
fun side_constraint_tac bnf constr_defs ctxt i =
let
val thms = constr_defs @ map mk_sym [rel_eq_of_bnf bnf, rel_conversep_of_bnf bnf,
rel_OO_of_bnf bnf]
in
(SELECT_GOAL (Local_Defs.unfold_tac ctxt thms) THEN' rtac (rel_mono_of_bnf bnf)
THEN_ALL_NEW atac) i
end
fun bi_constraint_tac constr_iff sided_constr_intros ctxt i =
(SELECT_GOAL (Local_Defs.unfold_tac ctxt [constr_iff]) THEN'
CONJ_WRAP' (fn thm => rtac thm THEN_ALL_NEW (REPEAT_DETERM o etac conjE THEN' atac)) sided_constr_intros) i
fun generate_relation_constraint_goal ctxt bnf constraint_def =
let
val constr_name = constraint_def |> prop_of |> HOLogic.dest_Trueprop |> fst o HOLogic.dest_eq
|> head_of |> fst o dest_Const
val live = live_of_bnf bnf
val (((As, Bs), Ds), ctxt) = ctxt
|> mk_TFrees live
||>> mk_TFrees live
||>> mk_TFrees (dead_of_bnf bnf)
val relator = mk_rel_of_bnf Ds As Bs bnf
val relsT = map2 mk_pred2T As Bs
val (args, ctxt) = Ctr_Sugar_Util.mk_Frees "R" relsT ctxt
val concl = HOLogic.mk_Trueprop (mk_relation_constraint constr_name (list_comb (relator, args)))
val assms = map (HOLogic.mk_Trueprop o (mk_relation_constraint constr_name)) args
val goal = Logic.list_implies (assms, concl)
in
(goal, ctxt)
end
fun prove_relation_side_constraint ctxt bnf constraint_def =
let
val old_ctxt = ctxt
val (goal, ctxt) = generate_relation_constraint_goal ctxt bnf constraint_def
val thm = Goal.prove ctxt [] [] goal
(fn {context = ctxt, prems = _} => side_constraint_tac bnf [constraint_def] ctxt 1)
in
Drule.zero_var_indexes (singleton (Variable.export ctxt old_ctxt) thm)
end
fun prove_relation_bi_constraint ctxt bnf constraint_def side_constraints =
let
val old_ctxt = ctxt
val (goal, ctxt) = generate_relation_constraint_goal ctxt bnf constraint_def
val thm = Goal.prove ctxt [] [] goal
(fn {context = ctxt, prems = _} => bi_constraint_tac constraint_def side_constraints ctxt 1)
in
Drule.zero_var_indexes (singleton (Variable.export ctxt old_ctxt) thm)
end
val defs = [("left_total_rel", @{thm left_total_alt_def}), ("right_total_rel", @{thm right_total_alt_def}),
("left_unique_rel", @{thm left_unique_alt_def}), ("right_unique_rel", @{thm right_unique_alt_def})]
fun prove_relation_constraints bnf lthy =
let
val transfer_attr = @{attributes [transfer_rule]}
val Tname = base_name_of_bnf bnf
fun qualify suffix = Binding.qualified true suffix Tname
val defs = map (apsnd (prove_relation_side_constraint lthy bnf)) defs
val bi_total = prove_relation_bi_constraint lthy bnf @{thm bi_total_alt_def}
[snd (nth defs 0), snd (nth defs 1)]
val bi_unique = prove_relation_bi_constraint lthy bnf @{thm bi_unique_alt_def}
[snd (nth defs 2), snd (nth defs 3)]
val defs = ("bi_total_rel", bi_total) :: ("bi_unique_rel", bi_unique) :: defs
val notes = maps (fn (name, thm) => [((qualify name, []), [([thm], transfer_attr)])]) defs
in
notes
end
(* relator_eq *)
fun relator_eq bnf =
[((Binding.empty, []), [([rel_eq_of_bnf bnf], @{attributes [relator_eq]})])]
(* predicator definition and Domainp and eq_onp theorem *)
fun define_pred bnf lthy =
let
fun mk_pred_name c = Binding.prefix_name "pred_" c
val live = live_of_bnf bnf
val Tname = base_name_of_bnf bnf
val ((As, Ds), lthy) = lthy
|> mk_TFrees live
||>> mk_TFrees (dead_of_bnf bnf)
val T = mk_T_of_bnf Ds As bnf
val sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf
val argTs = map mk_pred1T As
val args = mk_Frees_free "P" argTs lthy
val conjs = map (fn (set, arg) => mk_Ball (set $ Bound 0) arg) (sets ~~ args)
val rhs = Abs ("x", T, foldr1 HOLogic.mk_conj conjs)
val pred_name = mk_pred_name Tname
val headT = argTs ---> (T --> HOLogic.boolT)
val head = Free (Binding.name_of pred_name, headT)
val lhs = list_comb (head, args)
val def = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
val ((_, (_, pred_def)), lthy) = Specification.definition ((SOME (pred_name, SOME headT, NoSyn)),
((Binding.empty, []), def)) lthy
in
(pred_def, lthy)
end
fun Domainp_tac bnf pred_def ctxt i =
let
val n = live_of_bnf bnf
val set_map's = set_map_of_bnf bnf
in
EVERY' [rtac ext, SELECT_GOAL (Local_Defs.unfold_tac ctxt [@{thm Domainp.simps},
in_rel_of_bnf bnf, pred_def]), rtac iffI,
REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE], hyp_subst_tac ctxt,
CONJ_WRAP' (fn set_map => EVERY' [rtac ballI, dtac (set_map RS equalityD1 RS set_mp),
etac imageE, dtac set_rev_mp, atac, REPEAT_DETERM o eresolve_tac [CollectE, @{thm case_prodE}],
hyp_subst_tac ctxt, rtac @{thm iffD2[OF arg_cong2[of _ _ _ _ Domainp, OF refl fst_conv]]},
etac @{thm DomainPI}]) set_map's,
REPEAT_DETERM o etac conjE, REPEAT_DETERM o resolve_tac [exI, (refl RS conjI), rotate_prems 1 conjI],
rtac refl, rtac (box_equals OF [map_cong0_of_bnf bnf, map_comp_of_bnf bnf RS sym,
map_id_of_bnf bnf]),
REPEAT_DETERM_N n o (EVERY' [rtac @{thm box_equals[OF _ sym[OF o_apply] sym[OF id_apply]]},
rtac @{thm fst_conv}]), rtac CollectI,
CONJ_WRAP' (fn set_map => EVERY' [rtac (set_map RS @{thm ord_eq_le_trans}),
REPEAT_DETERM o resolve_tac [@{thm image_subsetI}, CollectI, @{thm case_prodI}],
dtac (rotate_prems 1 bspec), atac, etac @{thm DomainpE}, etac @{thm someI}]) set_map's
] i
end
fun prove_Domainp_rel ctxt bnf pred_def =
let
val live = live_of_bnf bnf
val old_ctxt = ctxt
val (((As, Bs), Ds), ctxt) = ctxt
|> mk_TFrees live
||>> mk_TFrees live
||>> mk_TFrees (dead_of_bnf bnf)
val relator = mk_rel_of_bnf Ds As Bs bnf
val relsT = map2 mk_pred2T As Bs
val T = mk_T_of_bnf Ds As bnf
val (args, ctxt) = Ctr_Sugar_Util.mk_Frees "R" relsT ctxt
val lhs = mk_Domainp (list_comb (relator, args))
val rhs = mk_pred pred_def (map mk_Domainp args) T
val goal = HOLogic.mk_eq (lhs, rhs) |> HOLogic.mk_Trueprop
val thm = Goal.prove ctxt [] [] goal
(fn {context = ctxt, prems = _} => Domainp_tac bnf pred_def ctxt 1)
in
Drule.zero_var_indexes (singleton (Variable.export ctxt old_ctxt) thm)
end
fun pred_eq_onp_tac bnf pred_def ctxt i =
(SELECT_GOAL (Local_Defs.unfold_tac ctxt [@{thm eq_onp_Grp},
@{thm Ball_Collect}, pred_def]) THEN' CONVERSION (subst_conv (map_id0_of_bnf bnf RS sym))
THEN' rtac (rel_Grp_of_bnf bnf)) i
fun prove_rel_eq_onp ctxt bnf pred_def =
let
val live = live_of_bnf bnf
val old_ctxt = ctxt
val ((As, Ds), ctxt) = ctxt
|> mk_TFrees live
||>> mk_TFrees (dead_of_bnf bnf)
val T = mk_T_of_bnf Ds As bnf
val argTs = map mk_pred1T As
val (args, ctxt) = mk_Frees "P" argTs ctxt
val relator = mk_rel_of_bnf Ds As As bnf
val lhs = list_comb (relator, map mk_eq_onp args)
val rhs = mk_eq_onp (mk_pred pred_def args T)
val goal = HOLogic.mk_eq (lhs, rhs) |> HOLogic.mk_Trueprop
val thm = Goal.prove ctxt [] [] goal
(fn {context = ctxt, prems = _} => pred_eq_onp_tac bnf pred_def ctxt 1)
in
Drule.zero_var_indexes (singleton (Variable.export ctxt old_ctxt) thm)
end
fun predicator bnf lthy =
let
val (pred_def, lthy) = define_pred bnf lthy
val pred_def = Morphism.thm (Local_Theory.target_morphism lthy) pred_def
val Domainp_rel = prove_Domainp_rel lthy bnf pred_def
val rel_eq_onp = prove_rel_eq_onp lthy bnf pred_def
fun qualify defname suffix = Binding.qualified true suffix defname
val Domainp_rel_thm_name = qualify (base_name_of_bnf bnf) "Domainp_rel"
val rel_eq_onp_thm_name = qualify (base_name_of_bnf bnf) "rel_eq_onp"
val rel_eq_onp_internal = Conv.fconv_rule (HOLogic.Trueprop_conv (Conv.arg1_conv
(Raw_Simplifier.rewrite lthy false @{thms eq_onp_top_eq_eq[symmetric, THEN eq_reflection]})))
rel_eq_onp
val pred_data = {rel_eq_onp = rel_eq_onp_internal}
val type_name = type_name_of_bnf bnf
val relator_domain_attr = @{attributes [relator_domain]}
val notes = [((Domainp_rel_thm_name, []), [([Domainp_rel], relator_domain_attr)]),
((rel_eq_onp_thm_name, []), [([rel_eq_onp], [])])]
val lthy = Local_Theory.declaration {syntax = false, pervasive = true}
(fn phi => Transfer.update_pred_data type_name (Transfer.morph_pred_data phi pred_data)) lthy
in
(notes, lthy)
end
(* BNF interpretation *)
fun transfer_bnf_interpretation bnf lthy =
let
val constr_notes = if dead_of_bnf bnf > 0 then []
else prove_relation_constraints bnf lthy
val relator_eq_notes = if dead_of_bnf bnf > 0 then []
else relator_eq bnf
val (pred_notes, lthy) = predicator bnf lthy
in
snd (Local_Theory.notes (constr_notes @ relator_eq_notes @ pred_notes) lthy)
end
val _ = Context.>> (Context.map_theory (bnf_interpretation
(bnf_only_type_ctr (fn bnf => map_local_theory (transfer_bnf_interpretation bnf)))))
(* simplification rules for the predicator *)
fun lookup_defined_pred_data lthy name =
case (Transfer.lookup_pred_data lthy name) of
SOME data => data
| NONE => (error "lookup_pred_data: something went utterly wrong")
fun prove_pred_inject lthy (fp_sugar:fp_sugar) =
let
val involved_types = distinct op= (
map type_name_of_bnf (#fp_nesting_bnfs fp_sugar)
@ map type_name_of_bnf (#live_nesting_bnfs fp_sugar)
@ map type_name_of_bnf (#bnfs (#fp_res fp_sugar)))
val eq_onps = map (Transfer.rel_eq_onp o lookup_defined_pred_data lthy) involved_types
val live = live_of_bnf (bnf_of_fp_sugar fp_sugar)
val old_lthy = lthy
val (As, lthy) = mk_TFrees live lthy
val predTs = map mk_pred1T As
val (preds, lthy) = mk_Frees "P" predTs lthy
val args = map mk_eq_onp preds
val cTs = map (SOME o certifyT lthy) (maps (replicate 2) As)
val cts = map (SOME o certify lthy) args
fun get_rhs thm = thm |> concl_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
fun is_eqn thm = can get_rhs thm
fun rel2pred_massage thm =
let
val live_step = @{lemma "x = y \<Longrightarrow> (eq_onp P a a \<and> x) = (P a \<and> y)" by (simp only: eq_onp_same_args)}
val kill_top1 = @{lemma "(top x \<and> P) = P" by blast}
val kill_top2 = @{lemma "(P \<and> top x) = P" by blast}
fun pred_eq_onp_conj conjs = List.foldr (fn (_, thm) => thm RS live_step)
@{thm refl[of True]} conjs
val conjuncts = if is_eqn thm then thm |> get_rhs |> HOLogic.dest_conj else []
val kill_top = Local_Defs.unfold lthy [kill_top2] #> Local_Defs.unfold lthy [kill_top1]
val kill_True = Local_Defs.unfold lthy [@{thm HOL.simp_thms(21)}]
in
thm
|> Drule.instantiate' cTs cts
|> Conv.fconv_rule (HOLogic.Trueprop_conv (Conv.arg_conv
(Raw_Simplifier.rewrite lthy false @{thms eq_onp_top_eq_eq[symmetric, THEN eq_reflection]})))
|> Local_Defs.unfold lthy eq_onps
|> (fn thm => if conjuncts <> [] then @{thm box_equals}
OF [thm, @{thm eq_onp_same_args}, pred_eq_onp_conj conjuncts |> kill_True]
else thm RS (@{thm eq_onp_same_args} RS iffD1))
|> kill_top
end
val rel_injects = #rel_injects fp_sugar
in
rel_injects
|> map (Local_Defs.unfold lthy [@{thm conj_assoc}])
|> map rel2pred_massage
|> Variable.export lthy old_lthy
|> map Drule.zero_var_indexes
end
(* fp_sugar interpretation *)
fun transfer_fp_sugar_interpretation fp_sugar lthy =
let
val pred_injects = prove_pred_inject lthy fp_sugar
fun qualify defname suffix = Binding.qualified true suffix defname
val pred_inject_thm_name = qualify (base_name_of_bnf (bnf_of_fp_sugar fp_sugar)) "pred_inject"
val simp_attrs = @{attributes [simp]}
in
snd (Local_Theory.note ((pred_inject_thm_name, simp_attrs), pred_injects) lthy)
end
val _ = Context.>> (Context.map_theory (fp_sugar_interpretation (fp_sugar_only_type_ctr
(fn fp_sugars => map_local_theory (fold transfer_fp_sugar_interpretation fp_sugars)))))
end